1. Field of the Invention
This invention relates to digital communication and, more particularly, to a digital matched filter in a digital baseband digital symbol detector.
2. Description of the Related Art
Quadriphase shift keying (QPSK) is a quadrature amplitude modulation (QAM) technique of phase modulating the digital information onto a carrier signal. QPSK communications systems are generally known in the art. In these systems, a transmitter determines the frequency and phase of the unmodulated carrier wave. The transmitter associate two bits of information into a symbol a.sub.k and encodes the symbol into one of four QPSK alphabet elements to form a complex-valued line-coded symbol s.sub.k where the subscript k indicates a sample index in a discrete-time sequence which can be written: ##EQU1## where s.sub.k =e.sup.j.theta.(k) and .theta.(k).epsilon.{.pi./4, 3.pi./4, 5.pi./4, 7.pi./4}.
Conversion of this discrete time sequence, which is defined only at instants t=kT, to the continuous time domain necessitates application of a filter with impulse response g(t), called the pulse shape. The output of the pulse shape filter is the convolution of {S.sub.k } with g(t) and is known as the baseband pulse-shaped information signal, s(t) ##EQU2##
In radio applications, s(t) is modulated on a complex sinusoidal carrier at a radio frequency in order to effectively radiate in the air medium. The modulation operation can be written mathematically as ##EQU3##
In practice, the operation is typically realized as ##EQU4##
Furthermore, the pulse shape filter g(t) is typically implemented as a unit pulse ##EQU5##
For this case, ##EQU6##
Therefore, for the duration of each successive symbol, s(t) remains a complex constant e.sup.j.theta.(k) and x(t) reduces to ##EQU7## where the continuous-time symbolphase .theta.(t) is defined as follows. ##EQU8##
It can be seen therefore that the transmitted signal is a real-valued sinusoid at the RF carrier frequency at one of four discrete information-bearing phases.
Demodulation at the receiver is mathematically described as ##EQU9## followed by a low-pass filter to eliminate the high-frequency products. The filter output is then y(t)=e.sup.j.theta.(k) for the duration of each successive symbol and the symbol sequence {a.sub.k } can be decoded therefrom.
Practical implementation of Equation 1 is made difficult by the requirement to multiply by a complex sinusoid of predetermined frequency and phase. Known as a recovered coherent carrier, this carrier must match the frequency and phase of the transmitted carrier. The requirement is relaxed in the technique of differential QPSK (DQPSK). In DQPSK, the transmitted data are differentially encoded, that is, they are represented by the difference in phase between two successive symbols. This differential encoding affects only the mapping of the symbols a.sub.k into line-coded symbols s.sub.k, by applying the revised mapping rule ##EQU10## in the development of the transmitter model above. The measured phase difference between any two successive receive symbols identifies the information element .theta.(k) regardless of any arbitrary fixed phase offset in the recovered carrier used for downconversion. Therefore, using the differential technique, the receiver does not need the absolute phase of the carrier to decode the transmitted symbols. In fact, small errors in the frequency of the recovered carrier can also be tolerated in such a system when it results in a phase shift with respect to the carrier which is small relative the size of .theta.(k).
Further technological difficulties with direct implementation of Equation 1 encourage a multi-step downconversion, rather than a single direct downconversion to baseband. The typical receiver first downconverts the modulated RF carrier to an intermediate frequency (IF) and then again to baseband. The first downconversion output x'(t) resulting from downconverting x(t) from its RF carrier to an intermediate frequency .omega..sub.IF can be written ##EQU11##
The IF signal x'(t) can be subsequently downconverted to baseband. For accurate detection, the frequencies used in the downconversion must be such that the net frequency shift due to downconversion operations closely approximates the transmitter RF carrier.
The absolute frequency of the RF carrier at the receiver input will vary due to time-varying conditions such as impedance changes in the transmitter oscillator load and temperature changes or aging affecting the oscillator's frequency. Therefore, oscillators used in the receiver for downconversion generally require some control to track these frequency variations. A circuit designed to perform these controls so as to accurately downconvert the signal is known in the art as a carrier recovery loop.
After downconversion to baseband, the first stage in a typical detector is a matched filter. The matched filter maximizes signal-to-noise power ratio (SNR) at its output for a given transmitted pulse shape. The maximization is optimally achieved when the impulse response of the matched filter is the mirror image (rotated on the t=0 axis) of the complex conjugate of the expected received symbol pulse shape, which is defined to be the transmitted pulse shape g(t) distorted by the communication channel. Thus the impulse response f(t) of an ideal matched filter can be given by the following equation: EQU f*(-t)=c.multidot.g(t)*b(t)
where b(t) represents the channel characteristics and c is an arbitrary constant. It is well-known in the art that this impulse response results in a filter with a maximum output SNR for any given pulse shape. In many circumstances, the channel characteristic can adequately be modeled by c.multidot.b(t)=1, and in the case of interest, g(t) is .PI.(t) which is real and symmetric about t=0, so f(t) can be reduced to EQU f(t)=.PI.(t).
The output of the downconversion from IF and subsequent filtering can be described by z(t): ##EQU12##
It is further well known in the art that after the filtering and downconversion operations, a symbol-rate sampler is conventionally used to translate the continuous time received signal into a discrete-time signal. When f(t) is the special case under consideration f(t)=.PI.(t), the convolution product z(t) and its discrete-time equivalent z(k) are related by: ##EQU13##
To complete the digital receiver a quantizing and decoding device (slicer) typically follows, converting the baseband, filtered, sampled received signal first to a line-coded symbol s.sub.k and then mapping s.sub.k to a two-bit binary symbol a.sub.k.
Equation 2 requires two multiplication operations, one for the real part of the integrand and one for the imaginary part of the integrand. Analog multipliers represent a technical manufacturing challenge and add some noise or distortion, resulting in performance loss. Therefore, a digital matched filter is desired which avoids the pitfalls associated with circuit designs implementing analog multipliers. A matched filter digitally implementing a multiplier function without a conventional analog multiplier would result in a considerably more repeatable fabrication.
Further, analog implementation of an integrator suffers from low tolerances in the manufacturing process; this is especially true in monolithic integrated circuits. An analog integrator in one circuit may have a very different time constant than an integrator in another circuit manufactured by the same process. Unlike integration performed by analog components on integrated circuits, digital integration is a precisely controllable function determined by circuit design rather than the physical features of its components. All digital integrators produced by the same process have essentially the same performance characteristics.
As digital integrators are more flexible than their analog counterparts, it is desirable to have a matched filter that performs integration through easily available digital circuits. While an analog integrator requires a selection of reference resistors and/or capacitors to provide multiple time constants, a digital integrator can be easily programmed to change its function. Hence, an all-digital implementation of a matched filter results in reduced system complexity, but improved performance and flexibility of operation.